STEP-1-2013 试题.pdf

Sixth Term Examination Papers 9465 MATHEMATICS 1 Morning TUESDAY 25 JUNE 2013 Time 3 hours Additional Materials Answer Booklet Formulae Booklet INSTRUCTIONS TO CANDIDATES Please read this page carefully but do not open this question paper until you are told that you may do so Write your name centre number and candidate number in the spaces on the answer booklet Begin each answer on a new page Write the numbers of the questions you answer in the order attempted on the front of the answer booklet INFORMATION FOR CANDIDATES Each question is marked out of 20 There is no restriction of choice All questions attempted will be marked Your final mark will be based on the six questions for which you gain the highest marks You are advised to concentrate on no more than six questions Little credit will be given for fragmentary answers You are provided with a Mathematical Formulae Booklet Calculators are not permitted Please wait to be told you may begin before turning this page This question paper consists of 7 printed pages and 1 blank page UCLES 2013 Section A Pure Mathematics 1 i Use the substitution x y where y 0 to fi nd the real root of the equation x 3 x 1 2 0 ii Find all real roots of the following equations a x 10 x 2 22 0 b x2 4x 2x2 8x 3 9 0 2In this question x denotes the greatest integer that is less than or equal to x so that 2 9 2 2 0 and 1 5 2 The function f is defi ned for x 0 by f x x x i Sketch the graph of y f x for 3 x 3 with x 0 ii By considering the line y 7 12 on your graph or otherwise solve the equation f x 7 12 Solve also the equations f x 17 24 and f x 4 3 iii Find the largest root of the equation f x 9 10 Give necessary and suffi cient conditions in the form of inequalities for the equation f x c to have exactly n roots where n 1 3 For any two points X and Y with position vectors x and y respectively X Y is defi ned to be the point with position vector x 1 y where is a fi xed number i If X and Y are distinct show that X Y and Y X are distinct unless takes a certain value which you should state ii Under what conditions are X Y Z and X Y Z distinct iii Show that for any points X Y and Z X Y Z X Z Y Z and obtain the corresponding result for X Y Z iv The points P1 P2 are defi ned by P1 X Y and for n 2 Pn Pn 1 Y Given that X and Y are distinct and that 0 0 1 4 0 tannx sec2xdx 1 n 1 and 1 4 0 secnx tanxdx 2 n 1 n ii Evaluate the following integrals 1 4 0 x sec4x tanxdxand 1 4 0 x2sec2x tanxdx 5The point P has coordinates x y which satisfy x2 y2 kxy 3x y 0 i Sketch the locus of P in the case k 0 giving the points of intersection with the coordinate axes ii By factorising 3x2 3y2 10 xy or otherwise sketch the locus of P in the case k 10 3 giving the points of intersection with the coordinate axes iii In the case k 2 let Q be the point obtained by rotating P clockwise about the origin by an angle so that the coordinates X Y of Q are given by X xcos y sin Y xsin y cos Show that for 45 the locus of Q is 2Y 2X 1 2 1 Hence or otherwise sketch the locus of P in the case k 2 giving the equation of the line of symmetry 6 By considering the coeffi cient of xrin the series for 1 x 1 x n or otherwise obtain the following relation between binomial coeffi cients n r n r 1 n 1 r 1 r n The sequence of numbers B0 B1 B2 is defi ned by B2m m j 0 2m j j andB2m 1 m k 0 2m 1 k k Show that Bn 2 Bn 1 Bn n 0 1 2 What is the relation between the sequence B0 B1 B2 and the Fibonacci sequence F0 F1 F2 defi ned by F0 0 F1 1 and Fn Fn 1 Fn 2for n 2 3 UCLES 2013 Turn over 7 i Use the substitution y ux where u is a function of x to show that the solution of the diff erential equation dy dx x y y x x 0 y 0 that satisfi es y 2 when x 1 is y x 4 2lnx x e 2 ii Use a substitution to fi nd the solution of the diff erential equation dy dx x y 2y x x 0 y 0 that satisfi es y 2 when x 1 iii Find the solution of the diff erential equation dy dx x2 y 2y x x 0 y 0 that satisfi es y 2 when x 1 8 i The functions a b c and d are defi ned by a x x2 x 0 c x 2x x d x x x 0 Write down the following composite functions giving the domain and range of each cb ab da ad ii The functions f and g are defi ned by f x x2 1 x 1 g x x2 1 x Determine the composite functions fg and gf giving the domain and range of each iii Sketch the graphs of the functions h and k defi ned by h x x x2 1 x 1 k x x x2 1 x 1 justifying the main features of the graphs and giving the equations of any asymptotes Determine the domain and range of the composite function kh 4 UCLES 2013 Section B Mechanics 9Two particles A and B are projected simultaneously towards each other from two points which are a distance d apart in a horizontal plane Particle A has mass m and is projected at speed u at angle above the horizontal Particle B has mass M and is projected at speed v at angle above the horizontal The trajectories of the two particles lie in the same vertical plane The particles collide directly when each is at its point of greatest height above the plane Given that both A and B return to their starting points and that momentum is conserved in the collision show that mcot M cot Show further that the collision occurs at a point which is a horizontal distance b from the point of projection of A where b Md m M and fi nd in terms of b and the height above the horizontal plane at which the collision occurs 10 Two parallel vertical barriers are fi xed a distance d apart on horizontal ice A small ice hockey puck moves on the ice backwards and forwards between the barriers in the direction perpendicular to the barriers colliding with each in turn The coeffi cient of friction between the puck and the ice is and the coeffi cient of restitution between the puck and each of the barriers is r The puck starts at one of the barriers moving with speed v towards the other barrier Show that v2 i 1 r 2v2 i 2r2 gd where viis the speed of the puck just after its ith collision The puck comes to rest against one of the barriers after traversing the gap between them n times In the case r 1 express n in terms of r and k where k v2 2 gd If r e 1 where e is the base of natural logarithms show that n 1 2 ln 1 k e2 1 Give an expression for n in the case r 1 5 UCLES 2013 Turn over 11 A B C 1 2 The diagram shows a small block C of weight W initially at rest on a rough horizontal surface The coeffi cient of friction between the block and the surface is Two light strings AC and BC are attached to the block making angles 1 2 and to the horizontal respectively The tensions in AC and BC are T sin and T cos respectively where 0 T sin show that the block will remain at rest provided W sin T cos where is the acute angle such that tan ii In the case W T tan where 2 show that the block will start to move in a direction that makes an angle with the horizontal 6 UCLES 2013 Section C Probability and Statistics 12 Each day I have to take k diff erent types of medicine one tablet of each The tablets are identical in appearance When I go on holiday for n days I put n tablets of each type in a container and on each day of the holiday I select k tablets at random from the container i In the case k 3 show that the probability that I will select one tablet of each type on the fi rst day of a three day holiday is 9 28 Write down the probability that I wi